Problem 87 - Calculus for Biology and Medicine

Question

Show that the function \(y=(1 / 2)^{x}\) can be written in the form \(y=e^{-\mu x}\), where \(\mu\) is a positive constant. Determine \(\mu .\)

Step-by-Step Solution

Answer

The function can be written as \(y = e^{-x \ln 2}\), where \(\mu = \ln 2\).

Step 1: Understand the problem

We need to express the given exponential function \(y = \left(\frac{1}{2}\right)^x\) in the form \(y = e^{-\mu x}\), where \(\mu\) is a positive constant. Our goal is to find the expression for \(\mu\).

Step 2: Convert the base to 'e'

The function \(y = \left(\frac{1}{2}\right)^x\) can be rewritten using the property of exponents that states \(a^x = e^{x \ln a}\). Here's how:1. Identify \(a = \frac{1}{2}\).2. Rewrite: \(\left(\frac{1}{2}\right)^x = e^{x \ln \left(\frac{1}{2}\right)}\).

Step 3: Simplify the exponent

Focus on the exponent of \(e\):- Calculate \(\ln \left(\frac{1}{2}\right) = \ln 1 - \ln 2 = 0 - \ln 2 = -\ln 2\).- Substitute back: \(e^{x \ln \left(\frac{1}{2}\right)} = e^{x (-\ln 2)} = e^{-x \ln 2}\).

Step 4: Write in the required form and identify \(\mu\)

Now we have the expression \(y = e^{-x \ln 2}\), which can be written in the form \(y = e^{-\mu x}\), where \(\mu = \ln 2\).Thus, \(\mu\) is the natural logarithm of 2.

Key Concepts

Properties of Exponents

Exponents are a fundamental concept in mathematics used to describe how many times a number, known as the base, is multiplied by itself. Understanding their properties is crucial when dealing with exponential functions. Here are some of the basic properties:

• **Multiplication:** The rule \( a^m \times a^n = a^{m+n} \) describes how to multiply powers with the same base by adding their exponents.
• **Division:** The property \( a^m \div a^n = a^{m-n} \) shows how to divide powers with the same base by subtracting the exponents.
• **Power of a Power:** According to this rule, \( (a^m)^n = a^{m\cdot n} \), meaning you multiply the exponents when raising a power to another power.
• **Zero Exponent:** Any non-zero base raised to the power of zero equals one: \( a^0 = 1 \).
• **Negative Exponent:** The rule \( a^{-n} = \frac{1}{a^n} \) suggests that a negative exponent indicates a reciprocal.

When addressing exponential functions such as \( y = (\frac{1}{2})^x \), we can leverage these properties to express them in forms involving the base of the natural logarithm \( e \). By recognizing the properties, we efficiently maneuver and simplify exponential expressions to fit different forms, aiding in problem-solving.

Natural Logarithm

The natural logarithm, denoted as \( \ln \), is a logarithmic function with the base \( e \), where \( e \approx 2.718 \). It's a critical function in calculus and real-world applications due to its unique mathematical properties. Here are some insights:

• **Definition:** The natural logarithm \( \ln(x) \) is the inverse of the exponential function \( e^x \). Therefore, if \( y = \ln(x) \), then \( e^y = x \).
• **Important Properties:**
  • \( \ln(1) = 0 \), since \( e^0 = 1 \).
  • \( \ln(e) = 1 \), as \( e^1 = e \).
  • \( \ln(ab) = \ln(a) + \ln(b) \), highlighting its additive property.
  • \( \ln(\frac{a}{b}) = \ln(a) - \ln(b) \), showcasing the logarithmic quotient rule.
  • \( \ln(a^b) = b \cdot \ln(a) \), emphasizing its scaling power.

In the context of rewriting exponential functions, the natural logarithm facilitates transitions between different expressions. For example, converting \( (\frac{1}{2})^x \) to an \( e \)-based expression involves \( \ln(\frac{1}{2}) \), enabling a smoother path from traditional exponents to natural logarithms for deeper comprehension.

Exponential Equations

Exponential equations are equations where variables appear as exponents. Solving these equations often involves steps that utilize the properties of exponents and logarithms to isolate the variables. Here's how they work:

• **Setting up:** Typically, these equations take the form \( a^x = b \). The goal is to find \( x \) by expressing both sides with the same base, or using logarithms if the same base cannot be easily achieved.
• **Utilizing Logarithms:** Applying logarithms on both sides is a powerful method:
  • Convert the exponential form to a logarithmic form: If \( a^x = b \), then \( x = \log_a(b) \).
  • Using \( \ln \): Particularly common is using the natural logarithm, as in \( x = \ln(b)/\ln(a) \).
• **Exponential Base 'e':** In scenarios where the base is \( e \), such as \( e^x = k \), solving becomes straightforward using \( x = \ln(k) \).

For solving exponential functions like \( y = (\frac{1}{2})^x \) as demonstrated in the task, understanding these methods helps in reformulating the equation in terms of \( e \), leading to efficient solutions and a deeper understanding of the broader applications.